I have a function
$$ f(x)=\left(1-e^{\lambda x}\right) \left(1+e^x \left(u-1-e^{-u x} u\right)-x\right)+\lambda \left(e^{\lambda x} (1-u)+u\right) \left(x+e^x \left(e^{-u x}-1+(u-1) x\right)\right) $$ where $\lambda > 0$, and $0<u<1$ are parameters, and $f(x)$ is defined on $[0,\infty)$.
Below is how the function looks like when I plot on the interval (0,3) with $\lambda = 0.5$ and $u=0.5$.
How to prove that $f(x)$ has exactly one real root on the interval $(0,\infty)$? The difficulty here is that $\lambda$ and $u$ are unknown parameters. I have no clue how to proceed. Any help is greatly appreciated.